for \(v \in \mathbb {R}^{t}\) . The TBT gives that \(\eta \)  is a diffeomorphism for sufficiently small \(\epsilon _1\) . Note that since \(\eta \)  is a diffeomorphism on a compact set, it is therefore also uniformly continuous. For ease of notation, assume that the parameterization is global, \(\psi ^{-1}_\theta = \psi ^{-1}\) . The general case is handled with a partition of unity, which is guaranteed to exist. Define the operator \(r_n^\triangle (u,v):= r(\eta (u, v)),\)  and let \(J(u,v)\)  be the Jacobian term for this transformation according to [1]}, which depends on the second fundamental form of the manifold and where \(J(u,v) = 1\)  when \(v= 0\) .
